A priori estimates and a blow-up criterion for the incompressible ideal MHD equations with surface tension and a closed free surface

We establish the a priori estimates and prove a blow-up criterion for the three-dimensional free boundary incompressible ideal magnetohydrodynamics equations. The fluid occupies a bounded region with a free boundary that is a closed surface, without assumptions of simple connectedness or periodicity of the region (thus, Fourier transforms cannot be applied), nor the graph assumption for the free boundary. The fluid is under the influence of surface tension, and flattening the boundaries using local coordinates is insufficient to resolve this problem. This is because local coordinates fail to preserve curvature, as the mean curvature of a flat boundary degenerates to zero. To address these challenges and circumvent the intricate issue of spatial regularity in Lagrangian coordinates, we utilize reference surfaces to represent the free boundary and develop new energy functionals that both preserve the material derivative and incorporate spatial-temporal scaling $\partial_t \sim \nabla^{\frac{3}{2}}$ in Eulerian coordinates. This method enables us to establish both low-order and high-order regularity estimates without any loss of regularity. More importantly, we prove a blow-up criterion and provide a complete classification of blow-ups, including the self-intersection of the free boundary (which the graph assumption cannot handle), the breakdown of the mean curvature, and the blow-up of the normal velocity (which Lagrangian coordinates fail to capture). To the best of our knowledge, this is the first result addressing the a priori estimates and the blow-up criterion for free boundary problems with surface tension in general regions.